The experimental analysis of resonators at their thermomechanical limits is hindered by resonance frequency fluctuations [1]} and the presence of measurement noise introduced by instrumentation. To illustrate the challenge to study the Allan deviation of micro- and nano-mechanical resonators in the absence of measurement noise, we note that to reach that condition the thermomechanical noise peak in Fig. REF (c) would have to exceed the background PSD at \(f<f_0\)  by a factor \(Q^2={9e8}{}\)  [2]}, which is much higher than the experimental ratio of 6.3. To overcome this problem, we ensure that the resonator operates in the regime where its response is dominated by white force noise by using the piezo to add a random force, with white spectral distribution and PSD \(S_0\) , to the sinusoidal actuation force provided by the PLL system. According to the fluctuation-dissipation theorem, the thermomechanical noise of a system is fully accounted by a force of such type and power spectral density (one-sided, on a per hertz basis) [3]}
\(S_{\rm thm}=4 c k_B T,\) 
